{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T21:31:08Z","timestamp":1791667868431,"version":"4.3.4"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T00:00:00Z","timestamp":1708041600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study presents a subclass S(\u03b2) of bi-univalent functions within the open unit disk region D. The objective of this class is to determine the bounds of the Hankel determinant of order 3, (\u2c673(1)). In this study, new constraints for the estimates of the third Hankel determinant for the class S(\u03b2) are presented, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. Here, we define these bi-univalent functions as S(\u03b2) and impose constraints on the coefficients \u2502an\u2502. Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses.<\/jats:p>","DOI":"10.3390\/sym16020239","type":"journal-article","created":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T03:37:27Z","timestamp":1708054647000},"page":"239","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["On Third Hankel Determinant for Certain Subclass of Bi-Univalent Functions"],"prefix":"10.3390","volume":"16","author":[{"given":"Qasim Ali","family":"Shakir","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Al-Qadisiyah, Diwaniyah 58001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7033-8993","authenticated-orcid":false,"given":"Waggas Galib","family":"Atshan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Al-Qadisiyah, Diwaniyah 58001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,16]]},"reference":[{"key":"ref_1","unstructured":"Duren, P.L. (1983). Grundlehren der Mathematischen Wissenschaften, Band 259, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1090\/S0002-9939-1967-0206255-1","article-title":"On a coefficient problem for bi-univalent functions","volume":"18","author":"Lewin","year":"1967","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_3","unstructured":"Brannan, D.A., and Clunie, J.G. (1979, January 1\u201320). Aspects of Contemporary Complex Analysis. Proceedings of the NATO Advanced Study Institute Held at the University of Durham, Durham, UK."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/BF00247676","article-title":"The minimal distance of the image boundary from the origin and the second coefficient of an univalent functions in: |z|<1","volume":"32","author":"Netanyahu","year":"1969","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_5","first-page":"77","article-title":"Some remarks on bi-univalent functions","volume":"39","author":"Kedzierawski","year":"1985","journal-title":"Ann. Univ. Mariae Curie Sklodowska Sect. A"},{"key":"ref_6","first-page":"70","article-title":"On some classes of bi-univalent functions","volume":"31","author":"Brannan","year":"1986","journal-title":"Stud. Univ. Babes Bolyai Math."},{"key":"ref_7","first-page":"867871","article-title":"Initial coefficient bounds for a general class of bi-univalent functions","volume":"2014","author":"Altinkaya","year":"2014","journal-title":"Int. J. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"032025","DOI":"10.1088\/1742-6596\/1294\/3\/032025","article-title":"Results on coefficients estimates for subclasses of analytic and bi-univalent functions","volume":"1294","author":"Atshan","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Atshan, W.G., Rahman, I.A.R., and Lupas, A.A. (2021). Some results of new subclasses for bi-univalent functions using quasi-subordination. Symmetry, 13.","DOI":"10.3390\/sym13091653"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1090\/S0002-9904-1963-10923-4","article-title":"Power series with integral coefficients","volume":"69","author":"Cantor","year":"1963","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_11","first-page":"186","article-title":"Initial Maclaurin coefficients bounds for new subclasses of bi-univalent functions","volume":"5","author":"Frasin","year":"2015","journal-title":"Theory Appl. Math. Comput. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6","DOI":"10.14419\/gjma.v5i1.6959","article-title":"Estimates on initial coefficients of certain subclasses of bi-univalent functions associated with quasi- subordination","volume":"5","author":"Patil","year":"2017","journal-title":"Glob. J. Math. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.aml.2010.05.009","article-title":"Certain subclasses analytic and bi-univalent functions","volume":"23","author":"Srivastava","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"155","DOI":"10.2298\/PIM2022155Y","article-title":"Coefficients assessment for certain subclasses of bi-univalent functions related with quasi-subordination","volume":"108","author":"Yalcin","year":"2020","journal-title":"Publ. L\u2019Institut Math. Nouv. S\u00e9r."},{"key":"ref_15","first-page":"337","article-title":"On the second Hankel determinant of a really mean p-valent functions","volume":"223","author":"Noonan","year":"1976","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_16","first-page":"1","article-title":"On H3(1) Hankel determinant for some classes of univalent functions","volume":"6","author":"Babalola","year":"2010","journal-title":"Inequal. Theory Appl."},{"key":"ref_17","first-page":"301","article-title":"Second Hankel determinant for bi-starlike and bi-convex functions of order \u03b1","volume":"271","author":"Deniz","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Orhan, H., \u00c7aglar, M., and Cotirla, L.-I. (2023). Third Hankel Determinant for a Subfamily of Holomorphic Functions Related with Lemniscate of Bernoulli. Mathematics, 11.","DOI":"10.3390\/math11051147"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"132","DOI":"10.30970\/ms.60.2.132-137","article-title":"Hankel and Toeplitz determinants for a subclass of analytic functions","volume":"60","author":"Buyankara","year":"2023","journal-title":"Mat. Stud."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"012044","DOI":"10.1088\/1742-6596\/1664\/1\/012044","article-title":"Second Hankel determinant for certain subclasses of bi-univalent functions","volume":"1664","author":"Atshan","year":"2020","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"349","DOI":"10.35378\/gujs.958309","article-title":"On the Hankel determinant of m-fold symmetric bi-univalent functions using a new operator","volume":"36","author":"Atshan","year":"2023","journal-title":"Gazi Univ. J. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1112\/jlms\/s1-8.2.85","article-title":"Eine Bemerkung \u00fcber ungerade Schlichte Funktionen","volume":"1","author":"Fekete","year":"1933","journal-title":"J. Lond. Math. Soc."},{"key":"ref_23","first-page":"4150210","article-title":"Coefficient bounds for subclasses of bi-univalent functions associated with the Chebyshev polynomials","volume":"2017","author":"Guney","year":"2017","journal-title":"J. Complex Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.18280\/mmep.100345","article-title":"On the third Hankel determinant of certain subclass of bi-univalent functions","volume":"10","author":"Darweesh","year":"2023","journal-title":"Math. Model. Eng. Probl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/s13370-021-00957-8","article-title":"New concept on fourth Hankel determinant of a certain subclass of analytic functions","volume":"33","author":"Rahman","year":"2022","journal-title":"Afr. Mat."},{"key":"ref_26","first-page":"4227493","article-title":"Bernardi Integral Operator and Its Application to the Fourth Hankel Determinant","volume":"2022","author":"Khan","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_27","unstructured":"Grenander, U., and Szeg\u00f6, G. (1958). Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences, University California Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/239\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:00:29Z","timestamp":1760104829000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/239"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,16]]},"references-count":27,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020239"],"URL":"https:\/\/doi.org\/10.3390\/sym16020239","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,16]]}}}